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Long-time behaviour of deterministic Mean Field Games with non-monotone interactions

Martino Bardi, Hicham Kouhkouh

Abstract

We consider deterministic Mean Field Games (MFG) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of 1st order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFG with cost functional satisfying various additional assumptions, but not the classical Lasry-Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFG some results of weak KAM theory.

Long-time behaviour of deterministic Mean Field Games with non-monotone interactions

Abstract

We consider deterministic Mean Field Games (MFG) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of 1st order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFG with cost functional satisfying various additional assumptions, but not the classical Lasry-Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFG some results of weak KAM theory.
Paper Structure (8 sections, 22 theorems, 129 equations)

This paper contains 8 sections, 22 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Static MFG
  3. Ergodic MFG
  4. Approximation by long-time asymptotics
  5. Convergence of $m^T$
  6. Convergence of $u^T$
  7. Stronger convergence for $\mathcal{A }$ singleton
  8. Appendix: On Mather measures

Key Result

Proposition 2.2

Under the assumptions contxF, contF, and Finfinity, there exists a solution to supp pb.

Theorems & Definitions (58)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 48 more